Chapter 2 Number Operation

Number Operation

Number operation(Arithmetic) is an operation describe what we need to do to numbers. There are four main operations which are addition(+), subtraction(-), multiplication(*) and division(/). 

Here, we will only focusing on binary and hexadecimal number system operations.

Binary number operation(Binary Arithmetic)
Addition 
Addition is the simplest number operation in binary number system.
Rule of binary addition,
0+0=0
0+1=1
1+0=1

1+1=0,and carries 1 to the next significant bit(most significant bit).
Note: The rules of the binary addition(without carries) are the same as the truth table of XOR gate.

Example for binary addition:
00010110+00001011

      1111   << carries
  00010110 =    22(base 10)
+00001011 =    11(base 10)
  00100001 =    33(base 10)

Subtraction
Rule of binary subtraction,
0-0=0
0-1=1,and borrow 1 from the next significant bit(most significant bit).
1-0=1
1-1=0

Example for binary subtraction:
01000100-00100010

    02     02   << borrows
  01000100 =    68(base 10) 

 -00100010 =   -34(base 10)

  00100010 =    34(base 10)

Subtracting a positive value are equivalent to adding a negative number which can represent by two's complement.

Multiplication

Multiplication in binary is similar to its decimal counterpart.
Rule of binary multiplication,
0x0=0
0x1=0
1x0=0
1x1=1,and no carry or borrow bits.

Let's 00101001 be A and 00000011 be B,
The number of A and B can be multiplied by partial products. For each digit in B, the product of A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used.
Since there are only two digits in binary, so there are only two possible outcomes of each partial multiplication:
1.If the digit of B is 0, then the partial product is 0.
2.If the digit of B is 1, then the partial product is A.
To get the final result of multiplication, you need to sum all of the partial product.


Example for binary multiplication,
00101001 x 00000101

    00101001 =   41(base 10)
  x00000101 =     5(base 10)
      1               carries
    00101001
  00000000
00101001             
    11001101 = 205(base 10)

Division

Division in binary is similar to its decimal counterpart.
01001100 / 00000100

           10011 = 19(base 10)
100|01001100 = 76/4(base 10)
        100
             110
             100
               100
               100


Hexadecimal number operation
Addition
In hexadecimal addition, the value will be carry into the next digit when the value exceed 1516.
Example,
AE86 + F3B6

  111           carries
    AE86 =   44678(base 10)
+  F3B6 =   62390(base 10)
  1A23C = 107068(base 10)

Subtraction
In hexadecimal subtraction, the value will be borrow from the most significant bit when then subtrahend value is bigger than minuend value.
Example,
F3B6 - AE86

    16         borrows
  F3B6 = 62390(base 10)
 -AE86 = 44678(base 10)
   4530 = 17712(base 10)

Multiplication

X	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
1	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
2	2	4	6	8	A	C	E	10	12	14	16	18	1A	1C	1E
3	3	6	9	C	F	12	15	18	1B	1E	21	24	27	2A	2D
4	4	8	C	10	14	18	1C	20	24	28	2C	30	34	38	3C
5	5	A	F	14	19	1E	23	28	2D	32	37	3C	41	46	4B
6	6	C	12	18	1E	24	2A	30	36	3C	42	48	4E	54	5A
7	7	E	15	1C	23	2A	31	38	3F	46	4D	54	5B	62	69
8	8	10	18	20	28	30	38	40	48	50	58	60	68	70	78
9	9	12	1B	24	2D	36	3F	48	51	5A	63	6C	75	7E	87
A	A	14	1E	28	32	3C	46	50	5A	64	6E	78	82	8C	96
B	B	16	21	2C	37	42	4D	58	63	6E	79	84	8F	9A	A5
C	C	18	24	30	3C	48	54	60	6C	78	84	90	9C	A8	B4
D	D	1A	27	34	41	4E	5B	68	75	82	8F	9C	A9	B6	C3
E	E	1C	2A	38	46	54	62	70	7E	8C	9A	A8	B6	C4	D2
F	F	1E	2D	3C	4B	5A	69	78	87	96	A5	B4	C3	D2	E1

Multiplication Table
Hexadecimal multiplication is similar to decimal multiplication.
Example,
8AC x 9A

  8AC  =  2220(base 10)
x   1A  =     26(base 10)
11     carries
56B8
8AC              
E178  =57720(base 10)

Division
Example,
CA8B / 2B

         4B6
2B/CA92
     AC    
     1E9
     1D9
        102
        102